Optimal. Leaf size=156 \[ -\frac{b d \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 e^2}+\frac{b d \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 e^2}+\frac{d \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^2}+\frac{a x}{e}+\frac{b \log \left (1-c^2 x^2\right )}{2 c e}+\frac{b x \tanh ^{-1}(c x)}{e} \]
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Rubi [A] time = 0.153831, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {5940, 5910, 260, 5920, 2402, 2315, 2447} \[ -\frac{b d \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 e^2}+\frac{b d \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 e^2}+\frac{d \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^2}+\frac{a x}{e}+\frac{b \log \left (1-c^2 x^2\right )}{2 c e}+\frac{b x \tanh ^{-1}(c x)}{e} \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5910
Rule 260
Rule 5920
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{d+e x} \, dx &=\int \left (\frac{a+b \tanh ^{-1}(c x)}{e}-\frac{d \left (a+b \tanh ^{-1}(c x)\right )}{e (d+e x)}\right ) \, dx\\ &=\frac{\int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{e}-\frac{d \int \frac{a+b \tanh ^{-1}(c x)}{d+e x} \, dx}{e}\\ &=\frac{a x}{e}+\frac{d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^2}-\frac{(b c d) \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{e^2}+\frac{(b c d) \int \frac{\log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{1-c^2 x^2} \, dx}{e^2}+\frac{b \int \tanh ^{-1}(c x) \, dx}{e}\\ &=\frac{a x}{e}+\frac{b x \tanh ^{-1}(c x)}{e}+\frac{d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^2}+\frac{b d \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^2}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )}{e^2}-\frac{(b c) \int \frac{x}{1-c^2 x^2} \, dx}{e}\\ &=\frac{a x}{e}+\frac{b x \tanh ^{-1}(c x)}{e}+\frac{d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^2}+\frac{b \log \left (1-c^2 x^2\right )}{2 c e}-\frac{b d \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 e^2}+\frac{b d \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^2}\\ \end{align*}
Mathematica [C] time = 2.44204, size = 315, normalized size = 2.02 \[ \frac{\frac{b \left (c d \text{PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-c d \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+e \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left (\frac{c d}{e}\right )}+\frac{1}{2} i \pi c d \log \left (1-c^2 x^2\right )+e \log \left (1-c^2 x^2\right )-2 c d \tanh ^{-1}(c x) \tanh ^{-1}\left (\frac{c d}{e}\right )-2 c d \tanh ^{-1}(c x) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-2 c d \tanh ^{-1}\left (\frac{c d}{e}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+2 c d \tanh ^{-1}\left (\frac{c d}{e}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )+c d \tanh ^{-1}(c x)^2-i \pi c d \tanh ^{-1}(c x)+2 c d \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+i \pi c d \log \left (e^{2 \tanh ^{-1}(c x)}+1\right )-e \tanh ^{-1}(c x)^2+2 c e x \tanh ^{-1}(c x)\right )}{c}-2 a d \log (d+e x)+2 a e x}{2 e^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.112, size = 217, normalized size = 1.4 \begin{align*}{\frac{ax}{e}}-{\frac{ad\ln \left ( cxe+cd \right ) }{{e}^{2}}}+{\frac{bx{\it Artanh} \left ( cx \right ) }{e}}-{\frac{b{\it Artanh} \left ( cx \right ) d\ln \left ( cxe+cd \right ) }{{e}^{2}}}+{\frac{bd\ln \left ( cxe+cd \right ) }{2\,{e}^{2}}\ln \left ({\frac{cxe+e}{-cd+e}} \right ) }+{\frac{bd}{2\,{e}^{2}}{\it dilog} \left ({\frac{cxe+e}{-cd+e}} \right ) }-{\frac{bd\ln \left ( cxe+cd \right ) }{2\,{e}^{2}}\ln \left ({\frac{cxe-e}{-cd-e}} \right ) }-{\frac{bd}{2\,{e}^{2}}{\it dilog} \left ({\frac{cxe-e}{-cd-e}} \right ) }+{\frac{b\ln \left ({c}^{2}{d}^{2}-2\, \left ( cxe+cd \right ) cd+ \left ( cxe+cd \right ) ^{2}-{e}^{2} \right ) }{2\,ce}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} a{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + \frac{1}{2} \, b \int \frac{x{\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x \operatorname{artanh}\left (c x\right ) + a x}{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + b \operatorname{atanh}{\left (c x \right )}\right )}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x\right ) + a\right )} x}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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